All the underlying graphs in our consideration are simple and connected, unless otherwise stated. A signed graph \(\Sigma =(G,\sigma )\)  consists of a underlying graph \(G=(V,E)\)  with a signature function \(\sigma :E\rightarrow \lbrace -1,1\rbrace \) .
The (unsigned) graph \(G\)  is said to be the underlying graph of \(\Sigma \) , while the function \(\sigma \)  is called the signature of \(\Sigma \) . In signed graphs, edge signs are usually interpreted as \(\pm 1\) . Signed graphs first appeared in works of Harary [1]} and Cartwright and
Harary [2]}, and the matroids of graphs were extended to matroids of signed graphs
by Zaslavsky [3]}. Chaiken [4]} and Zaslavsky [3]} obtained the Matrix-Tree Theorem for signed graph independently. In fact, the theory of signed graphs is a special case of that of gain graphs and of biased graphs [6]}. In the very beginnings, these graphs are studied in the context of social psychology where the vertices are considered as individuals, while positive edges represent friendships and negative edges enmities between them. The notion of balance was introduced by Harary in [1]}, it plays a central role in
the matroid theory of signed graphs. A signed cycle is called positive (resp. negative) if it contains an even (resp. odd) number of negative edges. A signed graph is balanced if all its cycles are positive; otherwise it is unbalanced. Unsigned graphs are treated as (balanced) signed graphs where all edges get a positive sign, that is, the all-positive signature.
